sensors Review Review on Carbon Nanomaterials-Based Nano-Mass and Nano-Force Sensors by Theoretical Analysis of Vibration Behavior Jin-Xing Shi 1, Xiao-Wen Lei 2 and Toshiaki Natsuki 3,4,* 1 Department of Production Systems Engineering and Sciences, Komatsu University, Nu 1-3 Shicyomachi, Komatsu, Ishikawa 923-8511, Japan; [email protected] 2 Department of Mechanical Engineering, University of Fukui, 3-9-1 Bunkyo, Fukui 910-8507, Japan; [email protected] 3 Faculty of Textile Science and Technology, Shinshu University, 3-15-1 Tokida, Ueda-shi 386-8567, Japan 4 Institute of Carbon Science and Technology, Shinshu University, 4-17-1 Wakasato, Nagano 380-8553, Japan * Correspondence: [email protected] Abstract: Carbon nanomaterials, such as carbon nanotubes (CNTs), graphene sheets (GSs), and carbyne, are an important new class of technological materials, and have been proposed as nano- mechanical sensors because of their extremely superior mechanical, thermal, and electrical perfor- mance. The present work reviews the recent studies of carbon nanomaterials-based nano-force and nano-mass sensors using mechanical analysis of vibration behavior. The mechanism of the two kinds of frequency-based nano sensors is firstly introduced with mathematical models and expressions. Af- terward, the modeling perspective of carbon nanomaterials using continuum mechanical approaches as well as the determination of their material properties matching with their continuum models are concluded. Moreover, we summarize the representative works of CNTs/GSs/carbyne-based Citation: Shi, J.-X.; Lei, X.-W.; nano-mass and nano-force sensors and overview the technology for future challenges. It is hoped Natsuki, T. Review on Carbon that the present review can provide an insight into the application of carbon nanomaterials-based Nanomaterials-Based Nano-Mass and nano-mechanical sensors. Showing remarkable results, carbon nanomaterials-based nano-mass and Nano-Force Sensors by Theoretical nano-force sensors perform with a much higher sensitivity than using other traditional materials as Analysis of Vibration Behavior. resonators, such as silicon and ZnO. Thus, more intensive investigations of carbon nanomaterials- Sensors 2021, 21, 1907. based nano sensors are preferred and expected. https://doi.org/10.3390/s21051907 Keywords: carbon nanotubes; carbyne; graphene sheets; nano-force sensor; nano-mass sensor; Academic Editor: Daniel Ramos theoretical analysis; vibration Received: 22 February 2021 Accepted: 5 March 2021 Published: 9 March 2021 1. Introduction Publisher’s Note: MDPI stays neutral During the last several decades, since the fast development of observation instruments with regard to jurisdictional claims in for nanotechnology such as scanning electron microscopy (SEM), transmission electron published maps and institutional affil- microscopy (TEM), scanning tunneling microscopy (STM), and atomic force microscopy iations. (AFM), a variety of carbon nanomaterials, e.g., fullerene [1], carbon nanotubes (CNTs) [2–4], graphene sheets (GSs) [5], and carbyne [6] were discovered (or predicted) and investigated by scholars (as shown in Figure1). For instance, according to the invention of STM, which won the Nobel Prize in Physics in 1986, Kroto et al. [1] first produced and observed a soccer ball-like C60 fullerene and were awarded the 1996 Nobel Prize in Chemistry, Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. and Iijima [3] synthesized and measured double-walled, five-walled, and seven-walled This article is an open access article CNTs with diameters of 5.5, 6.7 and 6.5 nm, respectively. CNTs with one wall are often distributed under the terms and called single-walled CNTs (SWCNTs), with two walls are named as double-walled CNTs conditions of the Creative Commons (DWCNTs), and with more than two walls are named as multi-walled CNTs (MWCNTs). Attribution (CC BY) license (https:// Using SEM and AFM, Novoselov et al. [5] first produced and observed single-layered creativecommons.org/licenses/by/ GSs. Similar to the naming of CNTs, GSs with one layer are often called single-layered 4.0/). GSs (SLGSs), with two layers are named as double-layered GSs (DLGSs), and with more Sensors 2021, 21, 1907. https://doi.org/10.3390/s21051907 https://www.mdpi.com/journal/sensors Sensors 2021, 21, 1907 2 of 20 than two layers are named as multi-layered GSs (MLGSs). The present work focuses on the carbon nanomaterials of CNTs and GSs, which are usually adopted as components of nanoelectromechanical systems (NEMS) because of their outstanding material properties. Figure 1. Kinds of carbon nanomaterials. Regarding the material properties of carbon nanomaterials, the representative one- dimensional (1D) CNTs and two-dimensional (2D) GSs exhibit extremely superior mechan- ical, thermal, electrical, and optical performance almost on the same level (e.g., [7–13]). For example, Shokrieh and Rafiee [7] concluded mechanical properties of CNTs determined from both theoretical and experiments methods, and indicated that Young’s modulus of CNTs could reach to the TPa range. Kumar et al. [8] made a review work of material prop- erties of GSs, where they summarized that GSs own Young’s modulus of 1 TPa, thermal conductivity of 1500~5000 W m−1K−1, electrical conductivity of 104 S/cm, and optical transmittance of 97.7%. Because of their exceptionally high electronic conductivities, the application of CNTs and GSs on transistors [9], nanoelectronics [10], and supercapaci- tor [11] were also reviewed by scholars. In addition, more detail of their thermal and optical properties could also refer to some of previous review works [12,13]. According to their outstanding material properties introduced above, CNTs and GSs have been proposed and applied as sensing elements in biosensors [14–16], strain sensors [17–19], and gas sen- sors [20–22]. Besides these three kinds of sensors, carbon nanomaterials are also expected to contribute to the fields of nano-mass and nano-force sensors, which are considered in the present work. When a mass is too tiny to be detected by a normal measure method, mass sensors using mechanical resonators belonging to micro-electromechanical systems (MEMS), even NEMS, need to be developed. It is well known that mechanical resonators can be used as inertial balances to detect tiny mass by measuring oscillation frequency shifts [23]. Abadal et al. [24] proposed a simple electromechanical model using polysilicon as a cantilevered resonator, which had the sensitivity to detect attogram scales (10−18 g) of mass. By means of a complementary metal-oxide-semiconductor (CMOS) circuitry, this model could calculate dynamic quantities of the current flowing through the resonator at the resonance frequency as well as static magnitudes of the collapse voltage and deflection of the resonator, so that the unknown mass could be detected from the feedback of the electrical specifications of the CMOS circuitry as well as the resonance frequency of the resonator. Using the similar approach, silicon-based mass sensors have been investigated and developed further [25–37]. However, silicon-based mass sensors have their limitations of mass detection due to their relatively lower material properties (e.g., Young’s modulus of 180 GPa [24]) and larger cross- section (e.g., thickness of 1 µm [24]) compared to carbon nanomaterials. Poncharal et al. [38] firstly produced a nano-mass sensor using a cantilevered MWCNT as the resonator and observed the electrically induced dynamic deflections of the resonator attached by a carbon particle. From calculating the resonance frequency as revealed by the deflected contours, they measured the mass of the attached carbon particle to be 22 ± 6 fg (1 fg = 10−15 g). After that, plenty of experimental and theoretical investigations of CNTs/GSs-based nano-mass Sensors 2021, 21, 1907 3 of 20 sensors were carried out (e.g., [39–48]), which owned much higher sensitivity (>10−21 g) than silicon-based mass sensors. Regarding force sensors, for manipulating nano particles, biomolecular or cells, develop- ment of high sensitivity force sensors with mechanical types [49–56], electrical types [57–64], and optical types [65–68], it has been a great challenge for advanced micro/nano-assembly and bio-engineering. Willemsen et al. [49] summarized the works of the detection of biomolecular interaction forces using AFM with silicon nitride probes by that time, in which the AFM probes were considered as force sensors that could detect the interaction forces between individual molecules in nN (10−9 N) range by mechanical evaluations, such as strain change or frequency shift. They pointed out that though AFM was a versatile and high enough instrument to discern individual molecules, it could only detect force in one direction, and it would be interesting to be able to measure lateral and torsional forces. Wang et al. [57] demonstrated a piezoelectric field effect transistor (i.e., a nano-force sensor) composed of a ZnO nanowire bridging across two electrodes, which could detect a force in nanonewton range acted on the nanowire by evaluating the decrease of conduc- tance. Hong et al. [66] developed a CNTs-based nano-force sensor composed a CNTs-based transistor suspended with dual-trap optical tweezers, which could detect external forces by monitoring the morphology changes of the transistor using three-dimensional (3D) scanning photocurrent microscopy. This developed nano-force sensor had ability to detect mechanical coupling between individual DNA molecules and the transistor in pN (10−12 N) range, which was much more sensitive than silicon nitride/ZnO-based nano-force sensors. More detail of the difference among the three types of force sensors can be found in two previous review works [69,70]. The present work mainly discusses the frequency-based nano-force sensors, i.e., using carbon nanomaterials as resonators for detecting tiny forces from the evaluation of the resonant frequency shifts [71]. Whether for nano-mass or nano-force sensors, the determination of the resonance frequency from the vibration analysis is very important work.